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Chapter 12: Problem 94

Find the points at which the plane \(a x+b y+c z=d\) intersects the \(x-y-\), and \(z\) -axes.

Short Answer

Expert verified
Answer: The points where the plane ax + by + cz = d intersects the x-axis, y-axis, and z-axis are, respectively, (d/a, 0, 0), (0, d/b, 0), and (0, 0, d/c).

Step by step solution

01

Intersection with the x-axis

To find the point at which the plane intersects the x-axis, we let y=0 and z=0. The equation of the plane becomes: \(ax + b(0) + c(0) = d\), which simplifies to: \(ax=d\). Now, solve for x: \(x=\frac{d}{a}\). The intersection point with the x-axis is then: \((\frac{d}{a}, 0, 0)\).
02

Intersection with the y-axis

To find the point at which the plane intersects the y-axis, we let x=0 and z=0. The equation of the plane becomes: \(a(0) + by + c(0) = d\), which simplifies to: \(by=d\). Now, solve for y: \(y=\frac{d}{b}\). The intersection point with the y-axis is then: \((0, \frac{d}{b}, 0)\).
03

Intersection with the z-axis

To find the point at which the equation of the plane intersects the z-axis, we let x=0 and y=0. The equation of the plane becomes: \(a(0) + b(0) + cz = d\), which simplifies to: \(cz=d\). Now, solve for z: \(z=\frac{d}{c}\). The intersection point with the z-axis is then: \((0, 0, \frac{d}{c})\). So, the points where the plane \(ax + by + cz = d\) intersects the \(x-y-\), and \(z\) -axes are, respectively, \((\frac{d}{a}, 0, 0)\), \((0, \frac{d}{b}, 0)\), and \((0, 0, \frac{d}{c})\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coordinate Geometry
Coordinate Geometry is an essential branch of mathematics that deals with points, lines, curves and shapes in two and three dimensions using coordinates. This system helps us to relate algebraic equations to geometrical figures through a set of coordinates, also known as Cartesian coordinates.

In the xy-plane, each point is defined by a pair of numbers, \(x, y\), which represent its horizontal and vertical distances from the origin. Similarly, in a 3-dimensional space, points are described using ordered triplets \(x, y, z\) which indicate how far along the x-, y-, and z-axis they are situated.

Coordinate Geometry also enables us to work with geometrical entities like straight lines and planes using equations, helping us to find intersections, distances, and angles. This mathematical field also serves as the backbone for analyzing the shapes via the algebraic path and technology applications, including computer graphics and engineering designs.
Equation of a Plane
In Coordinate Geometry, a plane can be represented by a mathematical equation in three-dimensional space. The general form of this equation is \(ax + by + cz = d\). Each plane is defined by parameters a, b, c, and d, which are constants that specify the orientation and position of the plane.

The coefficients \(a, b, \text{and}\ c\) determine the direction of the normal vector to the plane. This normal vector is perpendicular to every line lying entirely within the plane, providing us an effective way to gauge the plane's tilt relative to the axes.

Let's break down how this equation works:
  • To find intersections with the coordinate axes, we set two of the three variables to zero, allowing us to solve for the third.
  • For instance, setting y=0 and z=0 helps find the x-intercept, while setting x=0 and z=0 gives us the y-intercept.
  • This manipulation shows us how the plane slices through the 3D coordinate space, revealing the points at which it crosses each axis.
Understanding the equation of a plane provides a foundation to explore intersections, orientations, and sections of geometric figures within space.
3D Coordinate System
The 3D Coordinate System is an extension of the 2D Cartesian coordinate system, introducing an additional z-axis for better spatial representation. This system allows us to describe points, lines, and planes in a three-dimensional space through three coordinates: x, y, and z.

Here's how it works:
  • The x-axis usually runs left to right or right to left.
  • The y-axis typically runs up and down.
  • The z-axis introduces depth, running front to back or back to front.
Visualizing and understanding three axes allows us to better model and comprehend complex spatial relationships. This coordinate system expands the potential analysis from flat shapes to volumes and is extensively used in fields such as physics, engineering, computer graphics, and more.

With the 3D system, geometric problems like finding where a plane intersects an axis (as seen in the original exercise) become solvable by simplifying the algebraic representation of the structures involved. It transforms challenging spatial problems into structured algebraic equations, making them more approachable and solvable.

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Most popular questions from this chapter

Given three distinct noncollinear points \(A, B,\) and \(C\) in the plane, find the point \(P\) in the plane such that the sum of the distances \(|A P|+|B P|+|C P|\) is a minimum. Here is how to proceed with three points, assuming that the triangle formed by the three points has no angle greater than \(2 \pi / 3\left(120^{\circ}\right)\). a. Assume the coordinates of the three given points are \(A\left(x_{1}, y_{1}\right)\) \(\underline{B}\left(x_{2}, y_{2}\right),\) and \(C\left(x_{3}, y_{3}\right) .\) Let \(d_{1}(x, y)\) be the distance between \(A\left(x_{1}, y_{1}\right)\) and a variable point \(P(x, y) .\) Compute the gradient of \(d_{1}\) and show that it is a unit vector pointing along the line between the two points. b. Define \(d_{2}\) and \(d_{3}\) in a similar way and show that \(\nabla d_{2}\) and \(\nabla d_{3}\) are also unit vectors in the direction of the line between the two points. c. The goal is to minimize \(f(x, y)=d_{1}+d_{2}+d_{3} .\) Show that the condition \(f_{x}=f_{y}=0\) implies that \(\nabla d_{1}+\nabla d_{2}+\nabla d_{3}=0\) d. Explain why part (c) implies that the optimal point \(P\) has the property that the three line segments \(A P, B P,\) and \(C P\) all intersect symmetrically in angles of \(2 \pi / 3\) e. What is the optimal solution if one of the angles in the triangle is greater than \(2 \pi / 3\) (just draw a picture)? f. Estimate the Steiner point for the three points (0,0),(0,1) and (2,0).

Identify and briefly describe the surfaces defined by the following equations. $$x^{2}+y^{2}+4 z^{2}+2 x=0$$

What point on the plane \(x-y+z=2\) is closest to the point (1,1,1)\(?\)

Find the maximum value of \(x_{1}+x_{2}+x_{3}+x_{4}\) subject to the condition that \(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=16\).

Let \(w=f(x, y, z)=2 x+3 y+4 z\) which is defined for all \((x, y, z)\) in \(\mathbb{R}^{3}\). Suppose that we are interested in the partial derivative \(w_{x}\) on a subset of \(\mathbb{R}^{3}\), such as the plane \(P\) given by \(z=4 x-2 y .\) The point to be made is that the result is not unique unless we specify which variables are considered independent. a. We could proceed as follows. On the plane \(P\), consider \(x\) and \(y\) as the independent variables, which means \(z\) depends on \(x\) and \(y,\) so we write \(w=f(x, y, z(x, y)) .\) Differentiate with respect to \(x\) holding \(y\) fixed to show that \(\left(\frac{\partial w}{\partial x}\right)_{y}=18,\) where the subscript \(y\) indicates that \(y\) is held fixed. b. Alternatively, on the plane \(P,\) we could consider \(x\) and \(z\) as the independent variables, which means \(y\) depends on \(x\) and \(z,\) so we write \(w=f(x, y(x, z), z)\) and differentiate with respect to \(x\) holding \(z\) fixed. Show that \(\left(\frac{\partial w}{\partial x}\right)_{z}=8,\) where the subscript \(z\) indicates that \(z\) is held fixed. c. Make a sketch of the plane \(z=4 x-2 y\) and interpret the results of parts (a) and (b) geometrically. d. Repeat the arguments of parts (a) and (b) to find \(\left(\frac{\partial w}{\partial y}\right)_{x}\) \(\left(\frac{\partial w}{\partial y}\right)_{z},\left(\frac{\partial w}{\partial z}\right)_{x},\) and \(\left(\frac{\partial w}{\partial z}\right)_{y}\)
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